Trigonometry (Pure Mathematics 3)

Q: Do I need to memorise double-angle formulae for P3?

Yes, double-angle formulae are not provided in the exam formula booklet, so you must memorise all three forms (sin2A, cos2A, tan2A). Compound-angle formulae are given.

Q: What is the correct restricted range for arccos x?

The principal range for arccos x is [0, π] (or 0° to 180°). Always use this range when asked for principal values of inverse cosine, even if other angles produce the same cosine value.

Q: Can I use the t = tan(½θ) substitution for solving trig equations?

No, this substitution is explicitly not required for Edexcel IAL P3, and you should use R-form or double-angle methods instead to avoid errors and missing solutions.

Edexcel International A-Level Mathematics· P3 §2.1 to §2.3 (2018 specification Issue 3)· 25 min read

1. Reciprocal & Inverse Trigonometric Functions★★☆☆☆⏱ 6 min

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📘 Definition

Reciprocal Trigonometric Functions

$sec θ = \frac{1}{cos θ}, cosec θ = \frac{1}{sin θ}, cot θ = \frac{cos θ}{sin θ}$ These functions are undefined where their denominator equals zero.

Example:

$sec 6 0^{\circ} = \frac{1}{cos 6 0 ^{\circ}} = 2$

Inverse trigonometric functions are defined with restricted domains to make them one-to-one, so they return a unique principal value for each input. These ranges are not given in the formula booklet and must be memorized.

📘 Definition

Inverse Trigonometric Principal Ranges

$arcsin x$ : domain $[- 1, 1]$ , range $[- \frac{π}{2}, \frac{π}{2}]$ | $arccos x$ : domain $[- 1, 1]$ , range $[0, π]$ | $arctan x$ : domain $R$ , range $(- \frac{π}{2}, \frac{π}{2})$

📐 Worked Example

Find the exact value of $arccos (\frac{2}{2}) + arctan (1)$ in radians.

1
Evaluate $arccos (\frac{2}{2})$ : $cos (\frac{π}{4}) = \frac{2}{2}$ , and $\frac{π}{4}$ lies in the range $[0, π]$ , so $arccos (\frac{2}{2}) = \frac{π}{4}$
2
Evaluate $arctan (1)$ : $tan (\frac{π}{4}) = 1$ , and $\frac{π}{4}$ lies in the range $(- \frac{π}{2}, \frac{π}{2})$ , so $arctan (1) = \frac{π}{4}$
3
Sum the values: $\frac{π}{4} + \frac{π}{4} = \frac{π}{2}$

Exam tip:

Always verify that your principal value for inverse trig functions falls within the standard restricted range; values outside this range will not be awarded marks for principal value questions.

2. P3 Pythagorean Trigonometric Identities★★☆☆☆⏱ 5 min

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📘 Definition

P3 Pythagorean Identities

$sec^{2} θ = 1 + tan^{2} θ$ Derived by dividing $sin^{2} θ + cos^{2} θ = 1$ by $cos^{2} θ$ . $cosec^{2} θ = 1 + cot^{2} θ$ Derived by dividing $sin^{2} θ + cos^{2} θ = 1$ by $sin^{2} θ$ . Both identities must be memorized, they are not provided in the formula booklet.

📐 Worked Example

Prove that $(tan θ + cot θ) \equiv sec θ cosec θ$ .

1
Start with the more complex left-hand side (LHS) and rewrite using sin/cos terms:
$\text{LHS} = \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta}$
2
Combine the fractions over a common denominator:
$\text{LHS} = \frac{\sin^2 \theta + \cos^2 \theta}{\cos \theta \sin \theta}$
3
Use $sin^{2} θ + cos^{2} θ = 1$ , then rewrite the denominator as reciprocal functions:
$\text{LHS} = \frac{1}{\cos \theta \sin \theta} = \sec \theta \cosec \theta = \text{RHS}$
4
The identity is proven.

Exam tip:

When proving identities, always start with the more complex side and simplify to match the simpler side; avoid rearranging both sides of the ≡ sign unless explicitly doing identical operations to both.

3. Compound & Double-Angle Formulae★★★☆☆⏱ 7 min

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Compound-angle formulae for $sin (A \pm B)$ , $cos (A \pm B)$ and $tan (A \pm B)$ are provided in your exam formula booklet. Double-angle formulae are derived by substituting $B = A$ into the compound-angle formulae, and must be memorized as they are not given.

$sin 2 A = 2 sin A cos A$
$cos 2 A = cos^{2} A - sin^{2} A = 2 cos^{2} A - 1 = 1 - 2 sin^{2} A$
$tan 2 A = \frac{2 t a n A}{1 - t a n ^{2} A}$

Double-angle formulae can be rearranged for half-angle applications, e.g. $cos (\frac{θ}{2}) = \pm \frac{1 + c o s θ}{2}$ , where the sign depends on the quadrant of $\frac{θ}{2}$ .

📐 Worked Example

Prove that $cos x cos 2 x + sin x sin 2 x \equiv cos x$ .

1
Recognize the LHS matches the compound-angle formula for $cos (A - B) = cos A cos B + sin A sin B$ .
2
Let $A = 2 x$ and $B = x$ , substitute into the identity:
$\text{LHS} = \cos(2x - x) = \cos x = \text{RHS}$
3
The identity is proven.

Exam tip:

You may use sum-to-product identities (given in the formula booklet) if helpful for simplification, but they are not a required skill; always show every step of rearrangement for identity proofs to earn full method marks.

4. R-Form Transformations & Trigonometric Equations★★★★☆⏱ 7 min

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R-form is used to rewrite expressions of the form $a cos θ + b sin θ$ as a single trigonometric function, making equations much simpler to solve. R-form is not given in the formula booklet, so you must derive it explicitly from compound-angle formulae every time.

📘 Definition

R-form Derivation

To rewrite $a cos θ + b sin θ$ as $R cos (θ - α)$ (where $R > 0$ ): 1. Expand the target form: $R cos (θ - α) = R cos θ cos α + R sin θ sin α$ . 2. Equate coefficients: $R cos α = a$ , $R sin α = b$ . 3. Solve for $R = a^{2} + b^{2}$ and $tan α = \frac{b}{a}$ , selecting the correct quadrant for $α$ based on the signs of $a$ and $b$ .

📐 Worked Example

Rewrite $3 cos θ + 4 sin θ$ in the form $R cos (θ - α)$ , where $R > 0$ and $0^{\circ} < α < 9 0^{\circ}$ , giving $α$ to 1 decimal place. Hence solve $3 cos θ + 4 sin θ = 2$ for $0^{\circ} \leq θ \leq 36 0^{\circ}$ , giving solutions to 1 decimal place.

1
Expand $R cos (θ - α)$ and equate coefficients: $R cos α = 3$ , $R sin α = 4$
2
Calculate $R$ : $R = 3^{2} + 4^{2} = 5$
3
Calculate $α$ : $tan α = \frac{4}{3}$ , so $α = arctan (\frac{4}{3}) = 53. 1^{\circ}$ . The expression is $5 cos (θ - 53. 1^{\circ})$
4
Solve the equation: $5 cos (θ - 53. 1^{\circ}) = 2 ⟹ cos (θ - 53. 1^{\circ}) = 0.4$
5
Find all solutions for $θ - 53. 1^{\circ}$ in $- 53. 1^{\circ} \leq θ - 53. 1^{\circ} \leq 306. 9^{\circ}$ : principal value $66. 4^{\circ}$ , second solution $36 0^{\circ} - 66. 4^{\circ} = 293. 6^{\circ}$
6
Add $53. 1^{\circ}$ to both solutions: $θ = 66.4 + 53.1 = 119. 5^{\circ}$ , $θ = 293.6 + 53.1 = 346. 7^{\circ}$ , both in the required interval.

Exam tip:

Always adjust the interval for the shifted angle (e.g. $θ - α$ ) before solving, to avoid missing or extraneous solutions; verify all final solutions lie in the original interval given.

5. Common Pitfalls

Wrong move:

Using unrestricted ranges for inverse trig functions when calculating principal values.

Why:

Inverse trig functions are only one-to-one on their restricted domains, so values outside these ranges are not valid principal solutions.

Correct move:

Always use the standard restricted ranges: arcsin $[- π /2, π /2]$ , arccos $[0, π]$ , arctan $(- π /2, π /2)$ for principal value questions.

Wrong move:

Memorizing R-form as a pre-set formula instead of deriving it.

Why:

Different R-form variants (R sin vs R cos, ±α) have different coefficient rules, leading to sign errors if memorized incorrectly.

Correct move:

Always expand your target R-form, equate coefficients, and solve for R and α explicitly for every question.

Wrong move:

Forgetting that secθ/cosecθ are undefined when their denominator is zero.

Why:

This leads to extraneous solutions when solving equations involving reciprocal trig functions.

Correct move:

After solving an equation with sec/cosec/cot, check that none of your solutions make the original denominator zero, and discard any that do.

Wrong move:

Using the $t = tan (\frac{θ}{2})$ substitution to solve trig equations.

Why:

This substitution is explicitly not required for Edexcel IAL P3, and leads to wasted time, errors, and missing solutions where $tan (\frac{θ}{2})$ is undefined.

Correct move:

Use double-angle or R-form methods as specified in the syllabus to solve trig equations.

Wrong move:

Only providing one solution to trigonometric equations in a given interval.

Why:

Trigonometric functions are periodic, so there are almost always multiple solutions in a given interval.

Correct move:

After finding the principal solution, use the periodicity of the function (sin/cos period 360°/2π, tan period 180°/π) to find all valid solutions in the interval.

6. Quick Reference Cheatsheet

Concept	Formula / Rule	Memorise? (Y/N)
Reciprocal trig functions	$sec θ = 1/ cos θ$ , $cosec θ = 1/ sin θ$ , $cot θ = cos θ / sin θ$	Y
P3 Pythagorean identities	$sec^{2} θ = 1 + tan^{2} θ$ , $cosec^{2} θ = 1 + cot^{2} θ$	Y
Double-angle formulae	$sin 2 A = 2 sin A cos A$ , $cos 2 A = cos^{2} A - sin^{2} A$ , $tan 2 A = 2 tan A / (1 - tan^{2} A)$	Y
R-form calculation	$R = a^{2} + b^{2}$ , $tan α = \frac{coefficient of s i n θ}{coefficient of c o s θ}$ for $R cos (θ - α)$	Y
Compound-angle formulae	$sin (A \pm B) = sin A cos B \pm cos A sin B$ , $cos (A \pm B) = cos A cos B \mp sin A sin B$	N (given in booklet)
Inverse trig principal ranges	arcsin: $[- π /2, π /2]$ , arccos: $[0, π]$ , arctan: $(- π /2, π /2)$	Y

7. Frequently Asked

Do I need to memorise double-angle formulae for P3?

Yes, double-angle formulae are not provided in the exam formula booklet, so you must memorise all three forms (sin2A, cos2A, tan2A). Compound-angle formulae are given.

What is the correct restricted range for arccos x?

The principal range for arccos x is [0, π] (or 0° to 180°). Always use this range when asked for principal values of inverse cosine, even if other angles produce the same cosine value.

Can I use the t = tan(½θ) substitution for solving trig equations?

No, this substitution is explicitly not required for Edexcel IAL P3, and you should use R-form or double-angle methods instead to avoid errors and missing solutions.

Going deeper

official_documentEdexcel IAL Maths Formula BookletContains all given compound-angle identities for P3

What's Next

Now that you have mastered P3 trigonometry, you are ready to apply these skills to more advanced Pure Mathematics 3 topics, including differentiation and integration of trigonometric functions, which appear frequently on the WMA13 exam. Trigonometric identities are also essential for solving differential equations in P3 and for coordinate geometry problems involving parametric equations in later units. Make sure you practice identity proofs and R-form equation solving regularly, as these are high-mark questions that appear on almost every P3 paper. You should also review past paper questions to familiarize yourself with exam phrasing and common question structures for this topic, to maximize your score on exam day.